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A batter hits a baseball, NASCAR has a 21-car crash, or maybe Thor punches the Hulk. When one object collides with another, we can describe the interaction with an impact force. But putting a specific number on that force is actually quite difficult.Let’s go over some methods we can use to find an impact force.
Impact Force Estimation
Let's say a cart is rolling down a ramp and collides with a spring bumper, like so:
When the cart interacts with the spring bumper “wall,” there is a backwards-pushing force on the cart. We can call the direction of the ramp the x-direction. Newton's second law says:
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This force will cause the cart to accelerate—in this case, to the left, or the opposite way it was heading, which is why it slows down during impact and even reverses direction. But what is the magnitude of this force?
We can estimate this with a simple calculation and measurements of the cart’s velocity before and after impact. In this case, just for this starting example, I used a motion detector that can give the values for the velocity as the cart moves along the track. Right before the collision (v1), it was moving at 0.603 meters per second (to the right). After the collision (v2), it had a velocity of –0.572 m/s. (The minus sign means it's now moving in the opposite direction, to the left.) I can also get the impact time for this collision (Δt) at 0.325 seconds. Putting all of this together, I can now calculate the cart's acceleration during impact:
If I know the mass of the cart (which I do—it's 0.566 kilograms), then the product of the mass and acceleration gives the force. That means in this case the impact force is 2.04 newtons. But wait! That's actually not the impact force. That's the average impact force. If the collision pushes on the cart with a constant force of 2.04 newtons for 0.325 seconds, then it will have the same change in velocity as we see in the data. However, we don't know if it actually has a constant impact force or if it has a maximum peak and then a lower value for the rest of the impact. That means we might need another way to measure it.
Force Sensor
It might seem obvious that the best way to measure the force during impact is to … just actually measure the force. But how can you do that?A very simple method would be to use something like a spring scale. This is just a spring with some calibration markings on it. You can determine the force exerted on the spring by looking at the distance it is either compressed or stretched. However, this method won't be very practical for a collision that lasts less than a second. You wouldn’t be able to get an accurate reading during that short period of time.
As an alternative, there are digital methods for measuring force. A common method for a force probe is using an internal element that changes electrical resistance as it is stretched or compressed. That means you can just use a voltage measurement and calibrate it for force. The nice thing about voltage is that it can be measured many times per second, so you can see how the force value changes during the impact.
For the cart collision in my example, I also used a force probe to get data. Here's what the impact force looks like. (I should point out that I used a spring bumper during the collision to increase the impact time, so that we could more easily see how the force changes.)
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Now you can see that the force is not constant. At the beginning of the impact, the force increases up to a peak value and then decreases. (Yes, it's a negative value of force because of the way I set up the force probe. Don’t worry, it's fine.) In this case, the magnitude of the maximum impact force is 4.67 newtons, which is greater than the actual average value.
Imagine that the cart collided with something much stiffer than a spring bumper. A stiffer spring would exert a greater force over a shorter distance. With a greater force, the cart would stop and reverse directions in a shorter time. So in that case it’s possible the impact force could be much, much larger than 4 newtons.
Clearly, the force sensor is the best way to determine the impact force. You don't just get the maximum force, but you can see how long that maximum force is exerted. However, there is a problem with this method: You have to have a force sensor ready at the point of impact before the collision occurs. So if there is an unexpected crash, then you obviously won't be able to use this method.
Video Analysis
Collisions happen all the time in sports: Football players tackle each other, baseballs bounce off the wall (and sometimes the roof or even each other), and of course there is Bruce Lee’s famous 1-inch punch. Well, if you have a video, you can create a video analysis. The basic idea is to mark the location of an object in each frame of the video. If you know the size of each pixel in the video and the time between frames, known as the frame rate, you can get both position and time data for a moving object.
It just so happens that I recorded the motion of that cart crashing into the barrier. Even better, there's a meter stick right there in the scene so that I can determine the size of each pixel. With that, I get the following position-time data:
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Nothing amazing there—except that I didn't need any sensors to get the data. With this position-time data, I can look at any two position-time points and use them to calculate the velocity in the x-direction.
If I want an even better estimate for the velocity, I could use three position points. After that, I will have a list of velocity values for most of the time values. (It's fewer since I use three position points for each velocity point.) Since the acceleration is the rate of change of velocity, I can then use the velocity values to find acceleration values. Now I can create the following plot of acceleration as a function of time just from the video:
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Not only do I get the "shape" of the acceleration curve for the colliding cart, but I also get a maximum acceleration of –6.67 meters per second squared. With that acceleration and the mass of the cart (0.566 kilogram), we get a maximum impact force of 3.73 newtons.
This isn't quite the same value I obtained from the force sensor—and that's OK. There are two main reasons why the maximum force is different with this method. First, I've only collected a position point every 1/30th of a second, because my video frame rate was 30 frames per second. It's entirely possible to increase this frame rate, but I kept it at a normal value because that's what you usually see in real videos.
The second issue is that when I calculate the velocity and then acceleration, I'm actually finding an acceleration value for just a few points at a time. This can lead to some small errors that put the final value off a little bit.
Still, this video method is great. It's noninvasive, and you can do it after the fact. You really just need a video. (Knowing the mass of the object is helpful too.)
Accelerometer
If you don't want to use video analysis, there's another way to get the acceleration data (and then use that to find the force). It's possible to just measure the acceleration directly, and you probably have a device that can do it with you right now: It’s called a smartphone. Your phone has an accelerometer so that it can measure how it moves, and it’s used for things like lidar, augmented reality, and even long-exposure photos.
In my opinion, the best app that gives you acceleration data from your phone is PhyPhox. (It's free.) You can actually get data from all the sensors on your phone with this app, like pressure, magnetic field, and rotation.
But anyway, what happens if I stick my phone on the cart colliding with the barrier? Here's the data I get:
From this, I get a maximum acceleration of 6.55 m/s2. I can find the maximum impact force again using the mass of the cart (plus the iPhone, which increases the mass). This puts the largest force during impact at 5.32 newtons.
Of course, you can measure acceleration with sensors other than the ones on your phone. In fact, just about every modern car has some type of accelerometer that it uses to determine when to deploy the airbags: when it senses high accelerations during impact. You could also put other acceleration sensors in the car and measure the acceleration of different points in the vehicle. This would produce data that tells you how the car deforms, or moves relative to itself, during a crash.
Here's a fun—and safe—experiment you can try at home. Take your phone with the PhyPhox app and drop it from very short distances onto a pillow while measuring the acceleration. Next, try dropping the phone from the same height onto another soft object to compare the impact acceleration. Hopefully, you should find that the softer the landing target, the lower the value of acceleration (and thus force) during impact.
